SOLUTION: The radical {{{ sqrt (82 + sqrt (3360)) }}} can be expressed as {{{ sqrt (a) + sqrt (b) }}}, where a < b. What is the product ab?

Algebra ->  Radicals -> SOLUTION: The radical {{{ sqrt (82 + sqrt (3360)) }}} can be expressed as {{{ sqrt (a) + sqrt (b) }}}, where a < b. What is the product ab?      Log On


   

Question 1104114: The radical +sqrt+%2882+%2B+sqrt+%283360%29%29+ can be expressed as +sqrt+%28a%29+%2B+sqrt+%28b%29+, where a < b. What is the product ab?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


%28sqrt%28a%29%2Bsqrt%28b%29%29%5E2+=+a%2Bb%2B2%2Asqrt%28ab%29

So if sqrt%2882%2Bsqrt%283360%29%29+=+sqrt%28a%29%2Bsqrt%28b%29

we must have

a%2Bb=82 (1) and

2%2Asqrt%28ab%29+=+sqrt%283360%29
sqrt%28ab%29+=+sqrt%28840%29
ab+=+840 (2)

Since the problem only asked us to find the product ab, we are done.

We could go ahead and find the values of a and b by solving the pair of equations (1) and (2), using trial and error, or a graphing calculator, or algebra using either factoring or the quadratic formula; those values are 70 and 12.

So while the problem didn't ask us to evaluate the square root,

sqrt%2882%2Bsqrt%283360%29%29+=+sqrt%2812%29%2Bsqrt%2870%29

Answer by ikleyn(52777) About Me  (Show Source):
You can put this solution on YOUR website!
.
This problem was formulated INCORRECTLY.

The correct formulation is THIS: 

    The radical +sqrt+%2882+%2B+sqrt+%283360%29%29+ can be expressed as sqrt+%28a%29+%2B+sqrt+%28b%29, where a < b are positive integer numbers. 
    What is the product ab?

Then the solution by @greenestamps is correct.

Otherwise, without the requirement that "a" and "b" are integer, the problem has INFINITELY MANY solutions.